¶The 8. Theoreme. The 10. Proposition. If a number be partes of a number, and an other nūber the self same partes of an other number, then alternately what partes or part the first is of the third, the selfe same partes or part is the second of the fourth.
SVppose that the number AB be of the number C the selfe same partes, that an other number DE is of an other nūber F, and let AB be lesse then DE. Then I say, that alternately also what part or partes AB is of DE, the selfe same partes or part is C of F. Forasmuch as what partes AB is of C, the selfe same partes is DE of F: therefore how many partes of C there are
in
AB, so many partes of
F also are there in
DE. Deuide
AB into the partes of
C, that is, into
AG and
GB. And likewise
DE into the partes of
F, that is,
DH and
HE. Now then the multitude of these
AG and
GB, is equall vnto the multi∣tude of these
DH and
HE. And forasmuch as what part
AG is of
C, the selfe same part is
DH of
F, therefore alter∣nately also (by the former) what part or partes
AG is of
DH, the selfe same part or partes is
C of
F. And by the same reason also what part or partes
GB is of
HE, the same part or partes is
C of
F. Wherefore what part or partes
AG is of
DH, the selfe same part or partes is
AB of
DE (by the 6. of the seuenth). But what part or partes
AG is of
DH, the selfe same part or partes is it proued that
C is of
F. Wherefore what partes or part
AB is of D
E, the selfe same partes or part is
C of
F: which was required to be proued.